Open Mapping:
            A mapping  from one topological space  into another topological space  is said to be an open mapping if for every open set  in ,  is open in . In other words a mapping is open if it carries open sets over to open sets. A continuous mapping may not be open.

Example: Let  with topology  and let  with topology , then  defined by , ,  is continuous but not open, because  is open in  but  is not open in .

Closed Mapping:
             A mapping  from one topological space  into another topological space  is said to be an closed mapping if for every closed set  in ,  is closed in . In other words a mapping is closed if it carries closed sets over to closed sets.

Theorems:

• Let  and  be topological spaces. A bijective mapping  is open if and only if  is continuous.
• Let  and  be topological spaces. A bijective mapping  is closed if and only if  is continuous.
• Let  and  be topological spaces. A mapping  is open if and only if  for every open subset  of .